Combining 成（な）る (to become), and 可（べ）く (to be permissible), なるべく is an expression that literally means ‘as each opportunity becomes permissible/presents itself’. This is often simply translated as ‘as much as possible’, or ‘whenever possible’.
なるべく (usually written in hiragana) is regularly used before an entire phrase, to express that (A) should happen as much as possible.

• 外（そと）に出（で）かけるとき はなるべくドアのカギをかけてください。
  When you go out, lock your doors as often as possible.
• 仕事（しごと）で疲（つか）れていても、なるべく夕食（ゆうしょく）は食（た）べてください。
  Even if you are tired from work, please eat dinner whenever possible.

Due to なるべく expressing the ‘possibility’, or ‘having the opportunity’ to do something, it will sound considerably softer/more polite than できるだけ. できるだけ is a similar expression which more directly states that something will be done ‘to the full extent possible’.

• 明日（あした）はできるだけ早（はや）く来（き）てね。
  Tomorrow, please make sure to come as early as possible.
• 明日（あした）はなるべく早（はや）く来（き）てね。
  Tomorrow, please come as early as possible.